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Dislocation–disclination model of heterogeneous nucleation of HCP-martensite
Scripta Materialia 45 (2001) 939±945
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Dislocation±disclination model of heterogeneous nucleation of HCP-martensite M.Yu. Gutkin*,1, K.N. Mikaelyan1, and V.E. Verijenko School of Mechanical Engineering, University of Natal, Durban 4041, South Africa
Received 12 February 2001; accepted 7 July 2001
Abstract A dislocation±disclination model is proposed describing heterogeneous nucleation of an embryo of hcpmartensite at a tilt grain boundary segment containing some extrinsic dislocations. The corresponding energy gain is analysed in detail. The equilibrium and critical embryo sizes under external conditions of temperature and stress are examined and discussed. Ó 2001 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. Keywords: Heterogenous phase transformation; Martensite nucleation; Theory of defects; Grain boundaries; Dislocations
Introduction Martensitic fcc ! hcp phase transformation has extensively been investigated during the last 70 years. The mechanisms and conditions of its realisation have been studied by many authors on various materials such as Fe-based alloys and steels. Fe±Mn [1±9], Fe±Mn±Co [3,7], Fe±Mn±Cr [3], Fe±Mn±Mo [10], Fe±Mn±Ni [3], Fe±Mn±Si [6,11±14], Fe±Mn±Si±Cr±Ni [15,16], Fe±Cr±C [17], Fe±Cr±Ni [2,3,18±21]; pure Co [22±24] and Co-based alloys Co±Cr±Mo [25±27], Co±Fe [28], and Co±Ni [29,30]. In recent years, special attention has been paid to this martensitic transformation (MT) for its key role in shape memory [11±16], damping [5] and orthopaedic implant alloys [25±27]. On the other hand, the fcc ! hcp MT represents the simplest case among other MTs and may be considered as a good model example for better understanding the nature and properties of MTs. However, the number of appropriate theoretical models looks quite limited. Christian [22] suggested that the fcc ! hcp MT can be accomplished by the *
Corresponding author. Fax: +7-812-321-4771. E-mail address: [email protected] (M.Yu. Gutkin). 1 On leave from: Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bol'shoj 61, Vasilievskij Ostrov, St. Petersburg 199178, Russia.
1359-6462/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 1 1 5 - 0
940
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
Fig. 1. Model of hcp-martensite embryo nucleation at a tilt GB with EDs.
passage of Shockley partial dislocations (SPDs) on alternate f1 1 1g planes. The question of whether this process involves regular or irregular overlapping of stacking faults has been widely discussed by the group of investigators who studied speci®c dislocationbased models of MTs (for review, see Refs. [19,28]). The other group used classical thermodynamics to consider the problem of the nucleation and growth of a martensitic embryo (for review, see Refs. [31±34]). Olson and Cohen [31±33] did the most clear and comprehensive theoretical consideration of the initial steps of MTs. In the case of fcc ! hcp MT, their dislocation mechanism is similar to that given in Ref. [22]. They also proposed [31±33] the model of heterogeneous embryo nucleation at that site of a grain boundary (GB) where some extrinsic dislocations (EDs) exist (Fig. 1a). We propose below an approach similar to that by Olson and Cohen [31±33] but with a detailed description of elastic ®elds and energy of the hcp-embryo included in the analysis. Our model incorporates both dislocation and disclination terms. We analyse the total energy gain due to the hcp-embryo nucleation and show the existence of both the equilibrium and critical embryo sizes under external conditions of temperature and shear stress. The critical external stress which provides athermal embryo nucleation, is also studied in terms of the characteristic parameters of the GB where the MT takes place. Model Following the model developed in Refs. [31±33], consider a low-angle symmetric tilt GB modelled as an in®nite wall of straight edge intrinsic dislocations (IDs), which contains a number of lattice EDs stored during the previous treatment of the fcc-phase (Fig. 1a). Let the angle of misorientation across the GB and its period be x and h, respectively, and the spacing between the EDs l which is equal to double the interplanar distance between f1 1 1g planes in fcc-phase. The group of EDs together with the two closest IDs (all with equal Burgers vectors b) may be geometrically described as a dipole of wedge disclinations [35] having the arm 2a and strength X b=l (Fig. 1b). In a similar way, one can consider two other parts of the GB as a disclination dipole with the arm 2A and strength x b=h.
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
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Under an external shear stress s, each of the EDs and two closest IDs may split into two SPDs. In this case, one of these two SPDs remains at the GB while the other glides along the f1 1 1g plane and produces a stacking fault behind itself. As a result of the overlapping stacking faults, an embryo of hcp-martensite arises [31±33]. The ®nal model consists of two equivalent disclination dipoles having the strength X=2 and modelling two walls of edge-component SPDs, two walls of screw-component SPDs, and two rows of continuously distributed virtual edge dislocations modelling the eigenstrain e inside the hcp-embryo (Fig. 1c). Therefore, the elastic ®elds of the embryo are given by a superposition of elastic ®elds created by these dislocation and disclination con®gurations. The dierence in total free energies which are characteristic for the initial state of the system (Fig. 1a,b), W1 , and for its ®nale state (Fig. 1c), W2 , in analogy with [31±33], reads DW W2
W1 nq DGd DE 2cd;
1
where n is the number of atomic planes (in thickness) composing the fault, q the density of atoms in a close packed plane in moles per unit area, DG the chemical free energy dierence between parent and product phases, DE the strain energy gain, and c the free energy per unit area of the embryo/matrix interface, and d is the embryo size along the f1 1 1g plane. DG is de®ned as a molar quantity while DW and DE are determined per unit dislocation length. Let us calculate the term DE E2 E1 . The energy E1 of the defects shown in Fig. 1b is E1 EXs Exs EXint x EXc Exc ;
2
where EXs and Exs are the strain energies of the X- and x-dipoles, respectively, EXint x is the energy of interaction between the X- and x-dipoles, EXc and Exc are the sum core energies of dislocations which are modelled by the X- and x-dipoles, respectively. The energy E2 of the defect con®guration shown in Fig. 1c has a similar form, viz. s int Exs EX=2 E2 2EX=2 c Evintx Escr
X=2
Eext ;
int EX=2
x
c el 2EX=2 Exc Ev Escr 2EvintX=2
3
s int is the strain energy of a X=2-dipole, EX=2 where EX=2 X=2 the energy of interaction beint tween the X=2-dipoles, EX=2 x the energy of interaction of the X=2-dipoles with the xc the core energy of dislocations modelled by a X=2-dipole, Ev the strain dipole, EX=2 el the strain energy of screw-component SPDs, EvintX=2 energy of virtual dislocations, Escr int and Ev x are the energies of interaction of the virtual dislocations with the X=2- and xc is the core energy of screw-component SPDs, and Eext the dipoles, respectively, Escr work done by the external stress s to move the X=2-dipole to the distance d. Finally, DE reads
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M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
1 2 2 2 ~ ~ ~ ~ ~ DE=Da X Xfb~ 2sd=Dg d ln d 1 d ln 1 d 2 8 2 ~2 < d A~ 122 1 ~ ~ ~ A ln 4bs 1 m=l Xx : 2 A~ 122 9 2 ~2 2 1 2 1 2= ~ ~ ~ d A 2 d A 2 1 A~ ln d~2 ln 2 2 1 ~ 2 A 2 d~ A~ 122 ; " #( ) b~2s d~2 2 2 2 1 ~ ~ ~ ~ ln 1 d 4d cot d 2 e 1 m 2 d ln 1 d~2 l~
2
2
1 2
4e2 d~2 ln
d~2 ; 1 d~2
4
where D l=2p 1 m, l is the shear modulus, m the Poisson's ratio, bs the Burgers vector of the screw-component SPDs, and denotation ~z z=2a is used. Substitution of Eq. (4) into Eq. (1) gives the free energy gain DW due to the hcp-embryo nucleation at a GB under the stress s. Results and conclusion To analyse numerically the model suggested, the following values of parameters have been taken from Refs. [31±33] aspcharacteristic for p the n steel Fe±16Cr±13Ni: p p p8, q 3:9 10 5 mole m 2 , X 1= 6, l p 3, b p 2=2, bs p= 2 6, a p2 3, l 74 GPa, m 1=3, e 0:01. The size of a sample has been taken as p 3:6 A, R 0:01 m. Evidently, the model under consideration is highly sensitive to temperature T. To study the temperature eect, one can use data from Refs. [31±33] where the relations of the chemical free energy gain DG and free interface energy c with the temperature have been discussed in detail. Other values of the problem parameters have varied for dierent calculations. First, let us consider the contributions of dierent energy terms into the free energy gain given by Eq. (1), when s 0, A 3a, and the hcp-embryo size increases. In Fig. 2, the plots are shown which correspond to DE (curve 1), DW (2), 2cd (3), and nq DGd (4). The values DG 126 J mole 1 and c 0:01 J m 2 typical for the steel Fe±16Cr±13Ni at T 200 K [31±33], have been used in these calculations. When the embryo is small (Fig. 2a), DW < 0 and achieves its minimal value at d deq which may be considered as an equilibrium size of the embryo. This result is quite dierent from standard descriptions of MTs where there exist no equilibrium embryo sizes. The reason of this dierence is the strong interaction between the mobile defects describing the front of the growing embryo and the immobile defects located at the GB. In fact, when d~ < 10 (Fig. 2a), the energetics of the growing embryo is mainly determined by the elastic term DE. For larger embryo sizes, DW increases, achieves its maximal value at d dc which may
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
943
~ (a) small d, ~ (b) large d. ~ The plots represent DE (curve 1), DW (2), 2cd Fig. 2. Dierent terms of DW via the embryo size d: (3), and nq DGd (4).
be considered as a standard critical size of the martensitic embryo, and then decreases (Fig. 2b). This part of the curve DW d is quite typical of standard thermodynamic theories of MTs. The elastic term DE remains practically constant here and the energetics of the growing embryo is mainly determined by two other terms, the free interface and chemical energies. Next consider the eect of external stress s on the total energy gain DW (Fig. 3). When the embryo is small enough, its growth is mainly controlled by interaction of the x- and X=2-dipoles, and the eect of s is not too strong (Fig. 3a). As d increases, the in¯uence of s becomes stronger. For large d, s controls the behaviour of the curve DW d and greatly in¯uences (together with temperature) the characteristic embryo sizes. The equilibrium (critical) size increases (decreases) as s increases and T decreases (Fig. 4). Using Fig. 3, one can separate three dierent regimes of behaviour of DW d depending on s as follows. (i) Small s (s=l 0±0:002, curves 1 to 4); DW d < 0 for small and very large d, and DW d > 0 for intermediate d; DW achieves its minimal value at d deq , and its maximal value at d dc . There exists an energy barrier DW dc for the growing embryo. When s increases, deq increases while both dc and DW dc decrease. (ii) Intermediate s (s=l 0:003±0:005, curves 5 and 6); DW d < 0 for any d; other features are as in case (i). (iii) Large s (s=l P 0:007, curves 7 and 8); DW d < 0 for any d; no extreme values or energy barriers exist for the growing embryo. From the physical point
Fig. 3. The total free energy gain DW via the embryo size d~ for T 200 K and s=l 0 (1), 10 ~ (b) large d. ~ 3 10 3 (5), 5 10 3 (6), 7 10 3 (7), and 10 2 (8): (a) small d,
4
(2), 10
3
(3), 2 10
3
(4),
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M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
Fig. 4. The equilibrium (a) and critical (b) embryo sizes, d~eq and d~c , via the external shear stress s for dierent values of the temperature T 100 K (1), 200 K (2), 250 K (3), and 300 K (4).
of view, the boundary between regimes (ii) and (iii) is of special interest because it separates the thermo-activated regimes (i) and (ii) of embryo development from the athermal one (iii). One can introduce the corresponding critical shear stress sc which is determined as that minimal which provides the elimination of extreme points at the curve DW d, thus stimulating the athermal generation of a martensitic embryo. From Fig. 3, it is estimated as sc 0:007l, that gives 518 MPa for l 74 GPa. Numerical results indicate that the driving force F d o DW =od 0 is at the in¯exion point d din (where deq < din < dc ) when s sc . A similar equation was already used in Ref. [36]. Our calculations show that for all values of s and T, the characteristic point d din remains approximately the same, i.e., d~in 2:81. We can use this observation to estimate analytically the values of sc from F d din 0 that gives X 31:6 2A~ 12 1 X2 b2s nq DG 2c 2 1:4Xx ln : 5 1 m 2 sc e 2 ~ 2D 3 4 2Da l 31:6 2A 1 Dependencies of sc on the GB characteristics, A~ and x, are shown in Fig. 5 for ~ (Fig. 5a). Within our dierent T. It demonstrates the non-linear behaviour of sc A
Fig. 5. The critical external shear stress sc via the x-dipole parameters, A~ (a) and x (b), for the temperature values T 100, 200, 250, 275, 300, 350, and 400 K (from bottom to top).
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
945
model, where A~ is assumed to vary from 1 to 3, sc is positive and increases with increasing A~ reaching sc; max at A~ 3. The critical stress sc is negative for small values of x and positive for large values (Fig. 5b). It increases with increasing temperature T which is in accordance with expected physical behaviour.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
Parr, J. G. (1952). J Iron Steel Inst 171, 137. Otte, H. M. (1957). Acta Metall 5, 614. Cina, B. (1958). Acta Metall 6, 748. Remy, L. (1977). Metall Trans A 8A, 253. Jun, J.-H., & Choi, C.-S. (1999). J Mater Sci 34, 3421. Cotes, S., Guillermet, A. F., & Sade, M. (1999). Mater Sci Eng A 273±275, 503. Baruj, A., Guillermet, A. F., & Sade, M. (1999). Mater Sci Eng A 273±275, 507. Lee, Y.-K., & Choi, C.-S. (2000). Metall Mater Trans A 31, 355. Lu, X., Qin, Z., Zhang, Y., Ding, B., & Hu, Z. (2000). Scripta Mater 42, 433. Durlu, T. N. (1999). J Mater Sci 34, 2887. Sato, A., Yamaji, Y., & Mori, T. (1986). Acta Metall 34, 287. Jiang, B., & Qi, X. (1999). Bull Mater Sci (India) 22, 717. Guo, Z., Rong, Y., Chen, S., Hsu, T. Y. (Xu Zuyao), Hong, J., & Zhao, X. (1999). Mater Trans JIM 40, 193. Li, H., Dunne, D., & Kennon, N. (1999). Mater Sci Eng A 273±275, 517. Arruda, G. J., Buono, V. T. L., & Andrade, M. S. (1999). Mater Sci Eng A 273±275, 528. Li, J. C., Zhao, M., & Jiang, Q. (2000). Metall Mater Trans A 31, 581. Remy, L., & Pineau, A. (1977). Mater Sci Eng 28, 99. Lecroisey, F., & Pineau, A. (1972). Metall Trans 3, 387. Fujita, H., & Ueda, S. (1972). Acta Metall 20, 759. Brooks, J. W., Loretto, M. H., & Shallman, R. E. (1979). Acta Metall 27, 1828. Brooks, J. W., Loretto, M. H., & Shallman, R. E. (1979). Acta Metall 27, 1839. Christian, J. W. (1951). Proc Roy Soc A 206, 51. Houska, C. R., Averbach, B. L., & Cohen, M. (1960). Acta Metall 8, 81. Hitzenberger, C., Karnthaler, H. P., & Korner, A. (1985). Acta Metall 33, 1293. Huang, P., & L opez, H. F. (1999). Mater Lett 39, 244. Huang, P., & L opez, H. F. (1999). Mater Lett 39, 249. de J. Saldõvar Garcõa, A., Manõ Medrano, A., & Salinas Rodrõguez, A. (1999). Metall Mater Trans A 30, 1177. Mahajan, S., Green, M. L., & Brasen, D. (1977). Metall Trans A 8, 283. Hitzenberger, C., Karnthaler, H. P., & Korner, A. (1988). Acta Metall 36, 2719. Hayzelden, C., Chattopadhyay, K., Barry, J. C., & Cantor, B. (1991). Philos Mag A 63, 461. Olson, G. B., & Cohen, M. (1976). Metall Trans A 7, 1897. Olson, G. B., & Cohen, M. (1976). Metall Trans A 7, 1905. Olson, G. B., & Cohen, M. (1976). Metall Trans A 7, 1915. Roitburd, A. L. (1981). Sov Phys Dokl 26, 92. Romanov, A. E., & Vladimirov, V. I. (1992). Dislocations in Solids (p. 191). Amsterdam: North Holland. Chen, I.-W., & Chiao, Y.-H. (1985). Acta Metall 33, 1827.
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Dislocation±disclination model of heterogeneous nucleation of HCP-martensite M.Yu. Gutkin*,1, K.N. Mikaelyan1, and V.E. Verijenko School of Mechanical Engineering, University of Natal, Durban 4041, South Africa
Received 12 February 2001; accepted 7 July 2001
Abstract A dislocation±disclination model is proposed describing heterogeneous nucleation of an embryo of hcpmartensite at a tilt grain boundary segment containing some extrinsic dislocations. The corresponding energy gain is analysed in detail. The equilibrium and critical embryo sizes under external conditions of temperature and stress are examined and discussed. Ó 2001 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. Keywords: Heterogenous phase transformation; Martensite nucleation; Theory of defects; Grain boundaries; Dislocations
Introduction Martensitic fcc ! hcp phase transformation has extensively been investigated during the last 70 years. The mechanisms and conditions of its realisation have been studied by many authors on various materials such as Fe-based alloys and steels. Fe±Mn [1±9], Fe±Mn±Co [3,7], Fe±Mn±Cr [3], Fe±Mn±Mo [10], Fe±Mn±Ni [3], Fe±Mn±Si [6,11±14], Fe±Mn±Si±Cr±Ni [15,16], Fe±Cr±C [17], Fe±Cr±Ni [2,3,18±21]; pure Co [22±24] and Co-based alloys Co±Cr±Mo [25±27], Co±Fe [28], and Co±Ni [29,30]. In recent years, special attention has been paid to this martensitic transformation (MT) for its key role in shape memory [11±16], damping [5] and orthopaedic implant alloys [25±27]. On the other hand, the fcc ! hcp MT represents the simplest case among other MTs and may be considered as a good model example for better understanding the nature and properties of MTs. However, the number of appropriate theoretical models looks quite limited. Christian [22] suggested that the fcc ! hcp MT can be accomplished by the *
Corresponding author. Fax: +7-812-321-4771. E-mail address: [email protected] (M.Yu. Gutkin). 1 On leave from: Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bol'shoj 61, Vasilievskij Ostrov, St. Petersburg 199178, Russia.
1359-6462/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. on behalf of Acta Materialia Inc. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 1 1 5 - 0
940
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
Fig. 1. Model of hcp-martensite embryo nucleation at a tilt GB with EDs.
passage of Shockley partial dislocations (SPDs) on alternate f1 1 1g planes. The question of whether this process involves regular or irregular overlapping of stacking faults has been widely discussed by the group of investigators who studied speci®c dislocationbased models of MTs (for review, see Refs. [19,28]). The other group used classical thermodynamics to consider the problem of the nucleation and growth of a martensitic embryo (for review, see Refs. [31±34]). Olson and Cohen [31±33] did the most clear and comprehensive theoretical consideration of the initial steps of MTs. In the case of fcc ! hcp MT, their dislocation mechanism is similar to that given in Ref. [22]. They also proposed [31±33] the model of heterogeneous embryo nucleation at that site of a grain boundary (GB) where some extrinsic dislocations (EDs) exist (Fig. 1a). We propose below an approach similar to that by Olson and Cohen [31±33] but with a detailed description of elastic ®elds and energy of the hcp-embryo included in the analysis. Our model incorporates both dislocation and disclination terms. We analyse the total energy gain due to the hcp-embryo nucleation and show the existence of both the equilibrium and critical embryo sizes under external conditions of temperature and shear stress. The critical external stress which provides athermal embryo nucleation, is also studied in terms of the characteristic parameters of the GB where the MT takes place. Model Following the model developed in Refs. [31±33], consider a low-angle symmetric tilt GB modelled as an in®nite wall of straight edge intrinsic dislocations (IDs), which contains a number of lattice EDs stored during the previous treatment of the fcc-phase (Fig. 1a). Let the angle of misorientation across the GB and its period be x and h, respectively, and the spacing between the EDs l which is equal to double the interplanar distance between f1 1 1g planes in fcc-phase. The group of EDs together with the two closest IDs (all with equal Burgers vectors b) may be geometrically described as a dipole of wedge disclinations [35] having the arm 2a and strength X b=l (Fig. 1b). In a similar way, one can consider two other parts of the GB as a disclination dipole with the arm 2A and strength x b=h.
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
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Under an external shear stress s, each of the EDs and two closest IDs may split into two SPDs. In this case, one of these two SPDs remains at the GB while the other glides along the f1 1 1g plane and produces a stacking fault behind itself. As a result of the overlapping stacking faults, an embryo of hcp-martensite arises [31±33]. The ®nal model consists of two equivalent disclination dipoles having the strength X=2 and modelling two walls of edge-component SPDs, two walls of screw-component SPDs, and two rows of continuously distributed virtual edge dislocations modelling the eigenstrain e inside the hcp-embryo (Fig. 1c). Therefore, the elastic ®elds of the embryo are given by a superposition of elastic ®elds created by these dislocation and disclination con®gurations. The dierence in total free energies which are characteristic for the initial state of the system (Fig. 1a,b), W1 , and for its ®nale state (Fig. 1c), W2 , in analogy with [31±33], reads DW W2
W1 nq DGd DE 2cd;
1
where n is the number of atomic planes (in thickness) composing the fault, q the density of atoms in a close packed plane in moles per unit area, DG the chemical free energy dierence between parent and product phases, DE the strain energy gain, and c the free energy per unit area of the embryo/matrix interface, and d is the embryo size along the f1 1 1g plane. DG is de®ned as a molar quantity while DW and DE are determined per unit dislocation length. Let us calculate the term DE E2 E1 . The energy E1 of the defects shown in Fig. 1b is E1 EXs Exs EXint x EXc Exc ;
2
where EXs and Exs are the strain energies of the X- and x-dipoles, respectively, EXint x is the energy of interaction between the X- and x-dipoles, EXc and Exc are the sum core energies of dislocations which are modelled by the X- and x-dipoles, respectively. The energy E2 of the defect con®guration shown in Fig. 1c has a similar form, viz. s int Exs EX=2 E2 2EX=2 c Evintx Escr
X=2
Eext ;
int EX=2
x
c el 2EX=2 Exc Ev Escr 2EvintX=2
3
s int is the strain energy of a X=2-dipole, EX=2 where EX=2 X=2 the energy of interaction beint tween the X=2-dipoles, EX=2 x the energy of interaction of the X=2-dipoles with the xc the core energy of dislocations modelled by a X=2-dipole, Ev the strain dipole, EX=2 el the strain energy of screw-component SPDs, EvintX=2 energy of virtual dislocations, Escr int and Ev x are the energies of interaction of the virtual dislocations with the X=2- and xc is the core energy of screw-component SPDs, and Eext the dipoles, respectively, Escr work done by the external stress s to move the X=2-dipole to the distance d. Finally, DE reads
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M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
1 2 2 2 ~ ~ ~ ~ ~ DE=Da X Xfb~ 2sd=Dg d ln d 1 d ln 1 d 2 8 2 ~2 < d A~ 122 1 ~ ~ ~ A ln 4bs 1 m=l Xx : 2 A~ 122 9 2 ~2 2 1 2 1 2= ~ ~ ~ d A 2 d A 2 1 A~ ln d~2 ln 2 2 1 ~ 2 A 2 d~ A~ 122 ; " #( ) b~2s d~2 2 2 2 1 ~ ~ ~ ~ ln 1 d 4d cot d 2 e 1 m 2 d ln 1 d~2 l~
2
2
1 2
4e2 d~2 ln
d~2 ; 1 d~2
4
where D l=2p 1 m, l is the shear modulus, m the Poisson's ratio, bs the Burgers vector of the screw-component SPDs, and denotation ~z z=2a is used. Substitution of Eq. (4) into Eq. (1) gives the free energy gain DW due to the hcp-embryo nucleation at a GB under the stress s. Results and conclusion To analyse numerically the model suggested, the following values of parameters have been taken from Refs. [31±33] aspcharacteristic for p the n steel Fe±16Cr±13Ni: p p p8, q 3:9 10 5 mole m 2 , X 1= 6, l p 3, b p 2=2, bs p= 2 6, a p2 3, l 74 GPa, m 1=3, e 0:01. The size of a sample has been taken as p 3:6 A, R 0:01 m. Evidently, the model under consideration is highly sensitive to temperature T. To study the temperature eect, one can use data from Refs. [31±33] where the relations of the chemical free energy gain DG and free interface energy c with the temperature have been discussed in detail. Other values of the problem parameters have varied for dierent calculations. First, let us consider the contributions of dierent energy terms into the free energy gain given by Eq. (1), when s 0, A 3a, and the hcp-embryo size increases. In Fig. 2, the plots are shown which correspond to DE (curve 1), DW (2), 2cd (3), and nq DGd (4). The values DG 126 J mole 1 and c 0:01 J m 2 typical for the steel Fe±16Cr±13Ni at T 200 K [31±33], have been used in these calculations. When the embryo is small (Fig. 2a), DW < 0 and achieves its minimal value at d deq which may be considered as an equilibrium size of the embryo. This result is quite dierent from standard descriptions of MTs where there exist no equilibrium embryo sizes. The reason of this dierence is the strong interaction between the mobile defects describing the front of the growing embryo and the immobile defects located at the GB. In fact, when d~ < 10 (Fig. 2a), the energetics of the growing embryo is mainly determined by the elastic term DE. For larger embryo sizes, DW increases, achieves its maximal value at d dc which may
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
943
~ (a) small d, ~ (b) large d. ~ The plots represent DE (curve 1), DW (2), 2cd Fig. 2. Dierent terms of DW via the embryo size d: (3), and nq DGd (4).
be considered as a standard critical size of the martensitic embryo, and then decreases (Fig. 2b). This part of the curve DW d is quite typical of standard thermodynamic theories of MTs. The elastic term DE remains practically constant here and the energetics of the growing embryo is mainly determined by two other terms, the free interface and chemical energies. Next consider the eect of external stress s on the total energy gain DW (Fig. 3). When the embryo is small enough, its growth is mainly controlled by interaction of the x- and X=2-dipoles, and the eect of s is not too strong (Fig. 3a). As d increases, the in¯uence of s becomes stronger. For large d, s controls the behaviour of the curve DW d and greatly in¯uences (together with temperature) the characteristic embryo sizes. The equilibrium (critical) size increases (decreases) as s increases and T decreases (Fig. 4). Using Fig. 3, one can separate three dierent regimes of behaviour of DW d depending on s as follows. (i) Small s (s=l 0±0:002, curves 1 to 4); DW d < 0 for small and very large d, and DW d > 0 for intermediate d; DW achieves its minimal value at d deq , and its maximal value at d dc . There exists an energy barrier DW dc for the growing embryo. When s increases, deq increases while both dc and DW dc decrease. (ii) Intermediate s (s=l 0:003±0:005, curves 5 and 6); DW d < 0 for any d; other features are as in case (i). (iii) Large s (s=l P 0:007, curves 7 and 8); DW d < 0 for any d; no extreme values or energy barriers exist for the growing embryo. From the physical point
Fig. 3. The total free energy gain DW via the embryo size d~ for T 200 K and s=l 0 (1), 10 ~ (b) large d. ~ 3 10 3 (5), 5 10 3 (6), 7 10 3 (7), and 10 2 (8): (a) small d,
4
(2), 10
3
(3), 2 10
3
(4),
944
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
Fig. 4. The equilibrium (a) and critical (b) embryo sizes, d~eq and d~c , via the external shear stress s for dierent values of the temperature T 100 K (1), 200 K (2), 250 K (3), and 300 K (4).
of view, the boundary between regimes (ii) and (iii) is of special interest because it separates the thermo-activated regimes (i) and (ii) of embryo development from the athermal one (iii). One can introduce the corresponding critical shear stress sc which is determined as that minimal which provides the elimination of extreme points at the curve DW d, thus stimulating the athermal generation of a martensitic embryo. From Fig. 3, it is estimated as sc 0:007l, that gives 518 MPa for l 74 GPa. Numerical results indicate that the driving force F d o DW =od 0 is at the in¯exion point d din (where deq < din < dc ) when s sc . A similar equation was already used in Ref. [36]. Our calculations show that for all values of s and T, the characteristic point d din remains approximately the same, i.e., d~in 2:81. We can use this observation to estimate analytically the values of sc from F d din 0 that gives X 31:6 2A~ 12 1 X2 b2s nq DG 2c 2 1:4Xx ln : 5 1 m 2 sc e 2 ~ 2D 3 4 2Da l 31:6 2A 1 Dependencies of sc on the GB characteristics, A~ and x, are shown in Fig. 5 for ~ (Fig. 5a). Within our dierent T. It demonstrates the non-linear behaviour of sc A
Fig. 5. The critical external shear stress sc via the x-dipole parameters, A~ (a) and x (b), for the temperature values T 100, 200, 250, 275, 300, 350, and 400 K (from bottom to top).
M.Yu. Gutkin et al./Scripta Materialia 45 (2001) 939±945
945
model, where A~ is assumed to vary from 1 to 3, sc is positive and increases with increasing A~ reaching sc; max at A~ 3. The critical stress sc is negative for small values of x and positive for large values (Fig. 5b). It increases with increasing temperature T which is in accordance with expected physical behaviour.
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